Random time changes of Feller processes

TL;DR

This paper proves the existence and uniqueness of solutions for a class of stochastic differential equations driven by symmetric alpha-stable Lévy processes, using random time changes of Feller processes, and explores conditions for perpetual integrals to diverge.

Contribution

It introduces new methods for establishing the existence of Feller processes with unbounded coefficients via random time changes, extending previous results.

Findings

Unique weak solutions for SDEs with unbounded coefficients.

Conditions for random-time changes to produce conservative Feller processes.

New criteria for the divergence of perpetual integrals of Feller processes.

Abstract

We show that the SDE $dX_t = \sigma(X_{t-}) \, dL_t$, $X_0 \sim \mu$ driven by a one-dimensional symnmetric $\alpha$-stable L\'evy process $(L_t)_{t \geq 0}$, $\alpha \in (0,2]$, has a unique weak solution for any continuous function $\sigma: \mathbb{R} \to (0,\infty)$ which grows at most linearly. Our approach relies on random time changes of Feller processes. We study under which assumptions the random-time change of a Feller process is a conservative $C_b$-Feller process and prove the existence of a class of Feller processes with decomposable symbols. In particular, we establish new existence results for Feller processes with unbounded coefficients. As a by-product, we obtain a sufficient condition in terms of the symbol of a Feller process $(X_t)_{t \geq 0}$ for the perpetual integral $\int_{(0,\infty)} f(X_{s}) \, ds$ to be infinite almost surely.